I'm sure by now you are realizing that the wire gauge required is dependent on the load amperage you are trying to support. Until you have that nailed down, you are just guessing at what's required. You may want to consider going to 240 VAC, that is one way to get around the current dilemma (no pun intended).
The only way that will solve anything (wire gauge vs load) would be to use a 240--120 transformer at the far end, and the cost of that + weatherproof box just negated the wire savings
